Since the group E7_q is linear, it follows from Remark 4.1 of [1]} that \(a\) , \(b\) , \(c\) , \(d\) , \(e\) , \(f\) , \(g\)  must be non-negative integers.
Now if \(a+c=0\) , i.e., \(a=c=0\) , a direct check says that for any \(w\in W(\mathfrak {g}, \mathfrak {t}_f)^1\) , at least one coordinate of \(w\Lambda \)  in terms of the basis \(\lbrace \varpi _1, \dots , \varpi _7\rbrace \)  vanishes. Therefore,
\(\lbrace \mu -\rho _n^{(j)}\rbrace  + \rho _c =w \Lambda \) 
could not hold for any \(K\) -type \(\mu \) . This proves that \(a+c>0\) . Other inequalities can be similarly deduced.
